Środowiskowe Seminarium z Informacji i Technologii Kwantowych

The original and best-understood way of doing fault-tolerant quantum computing uses concatenated error-correction codes, with recent advances suggesting that they may be as good as more recent surface codes. Yet there are few tools to evaluate the qubit resources that concatenated codes require. Hence, we propose a method giving closed-form equations for the qubit resources required by any concatenated code. These equations remain simple for an arbitrary number of levels of concatenation, providing an ideal tool to compare and minimize the resource costs of such codes. We then evaluate the resources for gate operations that require the injection of complicated states called “magic states”. It was expected that such magic operations dominate the resources required by a calculation, and so numerous works proposing optimizations of their resource cost. Our work reveals that this expectation is wrong, as already seen in others’ work on surface codes; magic operations are rarely a dominant cost for concatenated codes. Optimizations affecting all operations naturally have more impact than those affecting magic operations alone, yet our unexpected conclusion is that the former can reduce qubit resources by a few orders of magnitude while the latter give only marginal reductions. More precisely, we show this in full detail for a 7-qubit scheme with Steane error-correction gadgets or flag-qubits gadgets, and argue that our observations are likely to be representative of the behavior for typical concatenated codes.